Title 



Linearized topologies and deformation theory
 
Author 


  
Abstract 



In this paper, for an underlying small category U endowed with a Grothendieck topology tau, and a linear category a which is graded over U in the sense of [13], we define a natural linear topology Ttau on a, which we call the linearized topology. Grothendieck categories in (noncommutative) algebraic geometry can often be realized as linear sheaf categories over linearized topologies. With the eye on deformation theory, it is important to obtain such realizations in which the linear category contains a restricted amount of algebraic information. We prove several results on the relation between refinement (eliminating both objects, and, more surprisingly, morphisms) of the nonlinear underlying site (U, tau), and refinement of the linearized site (a, Ttau). These results apply to several incarnations of (quasicoherent) sheaf categories, leading to a description of the infinitesimal deformation theory of these categories in the sense of [17] which is entirely controlled by the Gerstenhaber deformation theory of the small linear category a, and the Grothendieck topology tau on U. Our findings extend results from [17,12,7] and recover the examples from [21,20]. (C) 2015 Elsevier B.V. All rights reserved.   
Language 



English
 
Source (journal) 



Topology and its applications.  Amsterdam  
Publication 



Amsterdam : 2016
 
ISSN 



01668641
 
Volume/pages 



200(2016), p. 176211
 
ISI 



000370897400012
 
Full text (Publishers DOI) 


  
Full text (open access) 


  
